[A series of posts explaining a paper on the mathematical modeling of the spread of antiviral resistance. Links to other posts in the series by clicking tags, “Math model series” or “Antiviral model series” under Categories, left sidebar. Preliminary post here. Table of contents at end of this post.]

We are now almost through with the mathematical part of the model in the paper, “Antiviral resistance and the control of pandemic influenza,” by Lipsitch et al., published in *PLoS Medicine* (once **Methods** section is done, we will examine the **Results**, which are not mathematical but epidemiological). Our goal is to be able to follow each day how many people in the population move from being susceptible to infection with the influenza virus, to being infected with the virus, to “getting over” the infection (either by recovering or dying). We also want to keep track of how many are infected with resistant strains and allow for different transmission rates if you are infected with a virus sensitive to Tamiflu and received the drug, infected with a virus sensitive to Tamiflu but not received the drug, or infective with a virus resistant to Tamiflu (so treatment doesn’t matter because it doesn’t do anything). Recall that the number of susceptibles on a particular day is denoted by X, the number infected with sensitive virus and untreated, by Y_{SU}, the number infected with sensitive virus but treated, by Y_{ST}, the number infected with resistant virus by Y_{R} and the number removed, by Z. The transmission rate constants for each of these categories are the corresponding β_{SU}, β_{ST} and β_{R} (see previous posts for more explanation). Here’s the figure, again, to see this graphically:

How long does someone stay sick?

Suppose someone is in one of the Y-boxes. How many days do they stay there? Lipsitch et al. assume that duration of illness is “exponentially distributed with a mean duration of 1/v days” (last sentence of paragraph 2 in the **Methods** section). What does this mean? The exponential distribution is a common way to express time intervals between random events. The classic example is the lifetime of a lightbulb, which has an (assumed) fixed chance of burning out each day. The exponential distribution is commonly used for the length of illness of patients following the onset of a disease (here “burning out” corresponds to getting over the illness). Each day there is a certain probability a sick person will either get better or die (in either case move out of the “infected” category). So each day a fixed fraction of them will move out of the Y-boxes, although each day the number of people in the various Y-boxes changes. The computer has no problem keeping track of this.

What is the fixed fraction? While we don’t know it directly, we can still calculate it if we know the average lifetime of the lightbulb or the average length of illness. Lipsitch et al. use 3.3 days as the average length of illness for an untreated person infected with the sensitive strain. This allows them to calculate the v, in 1/v = 3.3 days and v determines the exact shape of the exponential distribution. This calculation comes from a mathematical fact that the average lifetime is equal to one divided by the rate (v) of the exponential distribution. This all works if you are willing to believe the exponential distribution is a good model for length of illness. Experience shows it is, at least to the extent that any errors are relatively minor.

In principle, treatment could shorten duration while infection with the resistant strain could lengthen it. The authors allow for different v_{T} and v_{R} in the figure for different daily recovery probabilities in these other cases (arrows leading out of the Y boxes to the Z box). However in their illustrative example they use identical values for all three, derived from an average illness length (3.3 days) extracted from their references [3], [4] and [5]. They note that the value of v doesn’t change the attack rate in their model structure. (See Table 1 of the paper.)

What proportion of the population will receive prophylaxis?

Next, we need to know the fraction of people susceptible to infection receiving prophylactic Tamiflu. This will depend on how much of the drug is available and policy decisions about how to apportion available stockpiles between those not yet sick (prophylaxis) and those who need it to treat infection (treatment). The fraction prophylaxed is designated f_{p}. The fraction of those sick and treated with Tamiflu is designated f_{T} (note that there is a small error in the paper at this point; it is misprinted as f_{p} in the first sentence of the 4th paragraph of **Methods**). For illustrative purposes, Lipsitch et al. chose 30% for both f_{p} and f_{T}, but the model allows different values to be chosen and this is discussed in the **Results** section. Initially they assume 30% of the susceptible population will receive prophylactic Tamiflu and 30% of those who become infected (and are infectious) will be treated with Tamiflu.

Where we’ve been and where we’re headed:

Now we have taken care of the two arrows leading out of the X box (representing the number of new cases arising daily and explained in the last post) and the three arrows from the Y-boxes to the Z box. In the next post we will finish up by taking care of the eight labeled arrows in the middle of the diagram. That will conclude the model itself. After that, we can move on to **Results**. The mathematics will be over. You needn’t have understood the **Methods** to understand the meaning of the **Results**. But it’s nice to know where they come from.

Table of contents for posts in the series:

The Introduction. What’s the paper about?

Sidebar: thinking mathematically

The rule book in equation form

Effects of treatment and prophylaxis on resistance

Effects of Tamiflu use and non drug interventions

Effects of fitness costs of resistance